Exercises
27
EXERCISES 1. For the given Lie group, find the corresponding infinitesimals X and Y. (i) x = cosh đ?œ€x + sinh đ?œ€y, y = cosh đ?œ€y + sinh đ?œ€x √ (ii) x = x2 − 2y + 2yeđ?œ€ , y = y eđ?œ€ , (iii) x = √
x 1+
2đ?œ€y2
,
y y= √ . 1 + 2đ?œ€y2
2. For the given infinitesimals X and Y , find the corresponding Lie group. (i) X = xy, Y = 1 (ii) X = x + y, Y = y, (iii) X = x2 − x, Y = y2 + y. 3. Find infinitesimal transformations leaving the following ODEs invariant. Use these to find a change of variables and reduce the original ODE to one that is separable and solve the equation. (i)
y e x2 dy = xy + + dx x xy
(ii)
dy 3y x5 = + dx x 2y + x3
(iii)
y + y3 dy = dx x + (x + 1)y2
Hints: Try: 1. X = A(x), Y = B(x)y, 2. X = A(x), Y = B(y), 3. X = A(y), Y = B(x). 4*. If we introduce the change of variables given in (2.34), namely X
đ?œ•r đ?œ•r +Y = 0, đ?œ•x đ?œ•y
đ?œ•s đ?œ•s +Y = 1, đ?œ•x đ?œ•y
such that the differential equation dy = F(x, y) dx